Four identical masses (each of mass M) are placed at the corners of a square of side a. The escape velocity of a body is from the ‘ centre of the quuare is:

To determine the escape velocity from the center of a square formed by four identical masses at its corners, we need to consider the gravitational forces acting on a mass placed at the center. The escape velocity is the minimum speed needed for an object to break free from the gravitational pull of the masses without any additional propulsion.

Understanding the Gravitational Force

Each mass at the corners of the square exerts a gravitational force on the mass located at the center. The gravitational force \( F \) between two masses is given by Newton's law of gravitation:

F = G \frac{m_1 m_2}{r^2}

Where:

• G is the gravitational constant (approximately \( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))
• m_1 and m_2 are the masses (in this case, one mass is \( M \) and the other is the mass at the center, which we can denote as \( m \))
• r is the distance between the two masses

Calculating the Distance

The distance from the center of the square to any corner is given by the formula:

r = \frac{a}{\sqrt{2}}

This is derived from the Pythagorean theorem, as the distance forms the hypotenuse of a right triangle with sides of length \( \frac{a}{2} \).

Net Gravitational Force at the Center

Since there are four identical masses, we can calculate the total gravitational force acting on the mass at the center. The force due to one mass is:

F_i = G \frac{M m}{(\frac{a}{\sqrt{2}})^2} = G \frac{M m}{\frac{a^2}{2}} = \frac{2G M m}{a^2}

Since there are four such masses, the total gravitational force \( F_{total} \) is:

F_{total} = 4 \cdot F_i = 4 \cdot \frac{2G M m}{a^2} = \frac{8G M m}{a^2}

Potential Energy at the Center

The gravitational potential energy \( U \) of the mass at the center due to the four masses is given by the formula:

U = -\sum \frac{G M m}{r} = -4 \cdot \frac{G M m}{\frac{a}{\sqrt{2}}} = -\frac{4 \sqrt{2} G M m}{a}

Escape Velocity Calculation

The escape velocity \( v_e \) can be derived from the principle of energy conservation. The kinetic energy \( K \) required to escape the gravitational field must equal the potential energy:

\frac{1}{2} m v_e^2 = -U

Substituting the expression for \( U \):

\frac{1}{2} m v_e^2 = \frac{4 \sqrt{2} G M m}{a}

We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):

\frac{1}{2} v_e^2 = \frac{4 \sqrt{2} G M}{a}

Multiplying both sides by 2 gives:

v_e^2 = \frac{8 \sqrt{2} G M}{a}

Taking the square root of both sides results in:

v_e = \sqrt{\frac{8 \sqrt{2} G M}{a}}

Final Expression for Escape Velocity

Thus, the escape velocity from the center of the square formed by four identical masses at its corners is:

v_e = \sqrt{\frac{8 \sqrt{2} G M}{a}}

This formula provides the necessary speed to overcome the gravitational attraction of the four masses and escape into space. Understanding this concept is crucial in fields like astrophysics and orbital mechanics, where gravitational interactions play a significant role.